A structure-aware construction of quantum BPX preconditioners for Q1 finite element discretizations of reaction-diffusion problems

We present a structure-aware construction of quantum BPX preconditioners for Q1 finite element discretizations of constant-coefficient reaction–diffusion equations on the unit cube, including the Poisson equation. Building on existing quantum finite element and finite difference methods, we express the preconditioned operator through explicit one-dimensional maps, tensor products, and norm-preserving interlevel transfers. Symmetric preconditioning gives multilevel coefficients, and its favorable conditioning alone does not ensure efficient recovery of a state in the original finite element nodal basis. We instead use the local $L^2$-orthonormal basis underlying the construction as the output representation of the same finite element function. Its coefficient norm equals the continuous $L^2$ norm. Assuming direct access to the preconditioned right-hand side, we prepare this orthonormal coefficient state with a represented function error at most $\epsilon$. The expected query complexity is $$\mathcal{O}\text{\hspace{0.17em}⁣}(d{L}^3\frac{\mathrm{\parallel }f{\mathrm{\parallel }}_{L^2}}{\mathrm{\parallel }u{\mathrm{\parallel }}_{L^2}}\log \text{\hspace{0.17em}⁣}(2+\frac{L\mathrm{\parallel }f{\mathrm{\parallel }}_{L^2}}{\epsilon }\left)\right),$$ where $u$ is the exact solution, $f$ is the right-hand side, $d$ is the spatial dimension, and $L={\log }_2(1\mathrm{/}h)$ for mesh width $h$. The bound holds for $0<\epsilon \le \mathrm{\parallel }u{\mathrm{\parallel }}_{L^2}\mathrm{/}2$ and depends only polylogarithmically on mesh refinement for fixed continuous data. We also analyze linear-functional estimation and illustrate the construction and continuous error bounds with a one-dimensional UnitaryLab experiment.